Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
709 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,4)(3,7,5,8)(9,10)(11,14)(12,17)(13,18)(15,19)(16,21)(20,22)(23,24)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(2,4)(3,5)(10,11)(12,19)(16,18)(20,24)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,3,6,5)(2,7,4,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,4,6,2)(3,7,5,8)(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(1,5,6,3)(2,7,4,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,38)(26,32,33,37)(27,35,28,36)(31,39,34,40)Polycyclic presentation:
2^8|a^2=gh,b^2=fh,c^2=f,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#58529, #8104193, #8104429, … |
| maximal normal subgroup of | C2×#58529, #8104193, #8104429, … |
| maximal quotient of | #585597, #585940, #587121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51517);